# Simple Definition

This is a non-mathematical, non-academic definition of continuous: you can draw your function without lifting your pencil.

## Discontinuous

## Continuous

# Actual Mathematical Definition

**very similar**$$\forall \epsilon > 0, \exists \delta > 0 \text{ s.t. if } ||x-c|| < \delta \to ||f(x) - f(c)|| < \epsilon$$ The only difference is the $0 <$ part. Why is that? Think about the first picture. If the x is super close (but not equal) to $x_0$, the f(x) values will approach the red point. It will work as expected.

Now, think about the continuous definition. If we consider a super small $\epsilon$, then if $x = x_0$, which less than any $\delta > 0$, the expected f(x) values and the jump point will be greater than epsilon.

# Not Continuous

# Examples

# A Function is Continuous

That means that the function is continuous at every point in its domain.

# Differentiability Implies Continuity

We will prove this statement in two steps. First, we show the Lipschitz Condition is true. Now, once we show that, we can show that it’s continuous.

## Lipschitz Condition

We know that f(x) is differentiable. That's a given. So, we begin with this $$\lim_{h \to 0} \frac{||f(p + h) - f(p) - Df(p)(h)||}{||h||} = 0$$ Because this is a limit, we can pick a $\delta$ such that the above fraction will be less than 1. $$\frac{||f(p + h) - f(p)||}{||h||} \leq \frac{||f(p + h) - f(p) - Df(p)(h)||}{||h||} + \frac{||Df(p)(h)||}{||h||} $$ (by the Triangle Inequality Theorem). $$\frac{||f(p + h) - f(p)||}{||h||} \lt 1 + \frac{||Df(p)(h)||}{||h||} $$ In the post about total derivatives, we prove not only that the derivative is a linear transformation but also that this inequality holds:$$\exists C>0 \text{ s.t. } ||T(x)|| \leq C||x||$$ We now write this $$\frac{||f(p + h) - f(p)||}{||h||} \lt 1 + \frac{C_1||h||}{||h||} $$ $$\frac{||f(p + h) - f(p)||}{||h||} \lt 1 + C_1$$ Now, we let C = $1 + C_1$ $$||f(p + h) - f(p)|| \leq \left(1 + C\right)||h||$$ Therefore, the Lipschitz Condition is satisfied.

## Proving Continuity from Lipschitz

# Lipschitz Continuity

In fact, if a function satisfies the Lipschitz condition, it is considered Lipschitz continuous. An easy way to see if something is Lipschitz continuous is if the derivative is bounded. Why is that? Well the Lipschitz condition states that you can create a line that is greater than or equal to the function. If the derivative is unbounded, then the function will exceed that line.